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OO nan _ n=lOO xn n-0...

Question

OO nan _ n=lOO xn n-0

OO nan _ n=l OO xn n-0



Answers

$a_{n}=\left(n^{1 / n}-1\right)^{n}$

Hello. So here we have the limit as N goes to infinity of one minus one over end to the end. Well, um This is just gonna reference a serum from the book. This is 3 10.4 point seven, which says that um the limit as N goes to infinity of one plus negative one to the end uh To the end is equal to E to the -1, so therefore disappointed to converge to E to the -1 or one over E.

When you're dealing with Bino meals, a really common symbol you might come across is this one. Let's pronounce n taking J at a time. And this symbol and taking jail time represents this formula. When you plug in different numbers to this equation, certain patterns start to emerge. For example, if you have and taking zero times, you plug that into the equation the B N factorial over zero factorial times and minus zero factorial. Now, if you work that out, you'll get n factorial over zero factorial, which is just one times and minus zero factorial, which is equivalent toe end factorial. Now you could see that these two ends will cancel out and they would just be left with one. Now if you take and taken one time, plug them in U N factorial over one factorial times and minus one factorial. If you work that out, you get n factorial over one factorial, which is one times and minus one factorial. Now the way factorial is work and factorial equals and times n minus one times and minus two and so on on and minus one factorial equals and minus one times and minus two and so on. So you see that the end minus two is cancel out the in mines ones cancel a and you're just left with n so and taking one times equals and

Okay, so let's use our definition for n chew through. So this is n factorial over in zero fact room times and minus Cyril, which is n factorial door and factories cancel and wait. We see that we're left with one. And now for n choose one, we have n factorial over one factorial times and minus one factorial. Well, we can rewrite end as and the times and minus one factorial. Oh, what a falling. One factor toys. This one. And then we can cancel this when we see that we're left with an And so we see that for any end chooses. So we get a one and for any and choose one, we get that end value.

All right. So for the question today, we need to figure out what n I need to figure out what n the binomial cushion of and zero is and the binomial coefficient of n one iss. So first thing we're gonna do is I'm going to put into the formula to try and, um uh, see if I can cancel anything out. So the 1st 1 and that over zero, it's more you'll and then and oh, sorry. And then n minus k, that toil. And that's the formula for binomial Corey Fischer. As we can see this, uh, zero factorial goes to one. Um, sorry. This case should actually be zero. So when we can't see everything out and kind of look at it from this way, we're actually gonna be left with and that over and factorial so in factory over in factorial is equal to one. So for every case, whenever there's a number and number as n and zero as R. K, we're going to get one as the answer. Not for this one. We have saying we're going to set up the same formula on then we have and minus one factorial. So the factorial of one is just one again, so we can kind of get rid of that. And now we're gonna be left with an factorial over and minus one factorial. Whenever you have something like n factorial over and minus one factorial, it's going like that. So I use a basic example. Say you have three factorial. Actually, I was gonna do this in a different color. Uh, say you have three factorial over two factorial that can really be written as so In this case, it's the same as N and then end minus one. It could be three times two terms. One, This one is two times one in this case, the to and the ones can't slow it. And we're just left with three. And the same thing is here. So all the ends before it are gonna cancel out and we're just gonna be left with. And so whenever there's a number over, um, other end and then one as R k, we're gonna be left with just the number. So if there was a seven for end, you just be left the seven. There's an eight there for end. You just be left with, um eight. So I hope this, uh, blow perf was helpful in understanding why these answers are with their


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